on a generalization of the bipartite graph d...
TRANSCRIPT
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
On a generalization of the bipartite graphD(k, q)
Tang YuanshengJoint work with Cheng Xiaoyan
School of Mathematical Sciences, Yangzhou University
NCHU, Aug. 2019
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Outline
1 Background
2 Generalized Graphs Γ(T, q)
3 Automorphisms of Γ(T, q)
4 Connectivity of Γ(T, q)
5 Projections and Lifts in Γ(T, q)
6 Girth of Γ(T, q)
7 Conclusion
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Simple Graphs
The graphs we consider in this talk are simple, i.e. undi-rected, without loops and multiple edges.
For a graph G, its vertex set and edge set are denoted byV (G) and E(G), respectively. The order of G is the numberof vertices in V (G).
The degree of a vertex v ∈ G is the number of the verticesthat are adjacent to it. A graph is said to be r-regular if thedegree of every vertex is equal to r.
An automorphism of G means a bijection φ from V (G) toitself such that {φ(v), φ(v′)} is an edge iff {v, v′} is.G is said to be edge-transitive if for any two edges {v1, v′1},{v2, v′2} there is an automorphism φ of G such that
{φ(v1), φ(v′1)} = {v2, v′2}.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Simple Graphs
A sequence v1v2 · · · vn of vertices of G is called a path oflength n if {vi, vi+1} ∈ E(G) for i = 1, 2, . . . , n− 1 and vj 6=vj+2 for j = 1, 2, . . . , n− 2.
A path v1v2 · · · vn is called a cycle further if its length n isnot smaller than 3 and v3v4 · · · vnv1v2 is still a path.
If G contains at least one cycle, then the girth of G, denotedby g(G), is the length of a shortest cycle in G.
In the literature, graphs with large girth and a high degreeof symmetry have been shown to be useful in different prob-lems in extremal graph theory, finite geometry, coding theory,cryptography, communication networks and quantum compu-tations.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Definition of D(k, q)
For prime power q and integer k ≥ 2, Lazebnik et al. in1995 proposed a bipartite graph, denoted by D(k, q), whichis q-regular, edge-transitive and of large girth.
The vertex sets L(k) and P (k) of D(k, q) are two copies ofFkq such that (l1, l2, . . . , lk) ∈ L(k) and (p1, p2, . . . , pk) ∈ P (k)are adjacent in D(k, q) iff
l2 + p2 = p1l1, (1)
l3 + p3 = p1l2, (2)
and, for 4 ≤ i ≤ k,
li + pi =
{pi−2l1, if i ≡ 0 or 1 (mod 4),p1li−2, if i ≡ 2 or 3 (mod 4).
(3)
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Girth of D(k, q)
In 1995 Lazebnik et al. showed some automorphisms ofD(k, q) and then proved g(D(k, q)) ≥ k + 4.
Since the girth of a bipartite graph must be even, we haveg(D(k, q)) ≥ k + 5 for odd k. In 1995 Furedi et al. proved
g(D(k, q)) = k + 5
for odd k with k+52 |(q − 1) and conjectured further
Conjecture A. g(D(k, q)) = k+5 for all odd k and all q ≥ 4.
This conjecture was proved to be valid:
in 2014 when k + 5 = 2ps for some s > 0;
in 2016 when k+52ps divides q − 1 for some s ≥ 0;
where p is the characteristic of the field Fq.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Connected Components of D(k, q)
The graph D(k, q) was shown to be disconnected in generalin 1996. The number of components of D(k, q) was deter-mined further for q ≥ 3 in 2004.
The connectivity and the lower bound of the girth ofD(k, q)imply that the components of D(k, q) provide the best-knownasymptotic lower bound for the greatest number of edges ingraphs of their order and girth. Indeed, at the present, thecomponents of D(k, q) provide the best general lower bound(for all but few exceptional values of t) on the Turan numbersex(n; {C3, . . . , C2t+1}) and ex(n; {C2t}).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Wenger Graphs
For n ≥ 1, Lazebnik and Viglione constructed in 2002 abipartite graph Gn(q) (indeed a generalization of the graphdefined by Wenger in 1991) whose vertex sets are L(n+1) andP (n + 1) such that two vertices (l1, l2, . . . , ln+1) ∈ L(n + 1)and (p1, p2, . . . , pn+1) ∈ P (n+1) are adjacent in Gn(q) if andonly if
li + pi = p1li−1, i = 2, 3, . . . , n+ 1. (4)
For n ≥ 3 and q ≥ 3, or n = 2 and q odd, the graph Gn(q)is semi-symmetric( edge-transitive but not vertex-transitive),connected when 1 ≤ n ≤ q− 1 and disconnected when n ≥ q,in which case it has qn−q+1 components, each isomorphic toGq−1(q).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Graphs Defined by Equation Systems
For n ≥ 2 and commutative ring R, Lazebnik and Woldarconstructed in 2001 a bipartite graphBΓn = BΓ(R; f2, . . . , fn)whose vertex sets Ln and Pn are two copies of Rn such that(l1, l2, . . . , ln) ∈ Ln and (p1, p2, . . . , pn) ∈ Pn are adjacent inBΓn if and only if
li + pi = fi(p1, l1, p2, l2, . . . , pi−1, li−1), i = 2, 3, . . . , n, (5)
where fi : R2i−2 → R are given functions.Some general properties of BΓn were exhibited by Lazebnik
and Woldar in that paper.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Monomial Graphs
When R is a finite field and the functions fi are mono-mials of p1 and l1, the graph BΓn is also called a monomialgraph. For positive integers k,m, k′,m′, it is proved in 2005that the monomial graphs BΓ2(Fq; pk1lm1 ) and BΓ2(Fq; pk
′1 l
m′1 )
are isomorphic if and only if {gcd(k, q − 1), gcd(m, q − 1)} ={gcd(k′, q − 1), gcd(m′, q − 1)} as multi-sets.
If q is odd, it was proved in 2007 that any monomial graphBΓ3 of girth ≥ 8 is isomorphic to a graph BΓ3(Fq; p1l1, pk1l2k1 )for k with (k, q) = 1. In particular, the integer k can berestricted to be 1 further if the odd prime power q is notgreater than 1010 or of form q = p2
a3b for odd prime p. Itwas further conjectured that:
Conjecture B. For any odd prime power q, every monomialgraph BΓ3 of girth ≥ 8 is isomorphic to BΓ3(Fq; p1l1, p1l21).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Two Conjectures on Permutation Polynomials
Let q be a power of an odd prime p and 1 ≤ k ≤ k − 1.
Conjection C. Xk[(X + 1)k − xk] ∈ Fq[X] is a permutationpolynomial if and only if k is a power of p.
Conjection D. [(X + 1)2k − 1]Xq−1−k − 2Xq−1 ∈ Fq[X] is apermutation polynomial if and only if k is a power of p.
It was proved in 2007 that Conjecture B is true if Conjec-tures C and D are true.
In 2019, Xiangdong Hou et.al proved that Conjecture B istrue though the Conjectures C and D are still open.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Condition for the Index Sets
In this talk, T denotes a set of binary sequences satisfying
C1: T contains the null sequence η and the sequences ob-tained from those in T \ {η} by deleting the last bits.
We note that T corresponds indeed a binary tree.
Let L(T ) and R(T ) be two copies of F|T |+1q , where |T | is
the size of T .We will denote the vectors in L(T ) by [l] (or in R(T ) by〈r〉, respectively). The entries of vectors in L(T ) ∪ R(T ) areindexed by the elements in T ∪ {∗}, where ∗ is a symbol notin T and used to index the colors of the vectors.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Definition of the Generalized Graphs
Let Γ(T, q) be the bipartite graph with vertex sets L(T ),R(T ) such that [l] ∈ L(T ) and 〈r〉 ∈ R(T ) are adjacent inΓ(T, q) if and only if
lη + rη = l∗r∗, (6)
and
lα0 + rα0 = r∗lα, for α0 ∈ T, (7)
lβ1 + rβ1 = l∗rβ, for β1 ∈ T. (8)
If we define ∗0σ = ∗1σ = σ for any σ ∈ {0, 1}∗, the equa-tion (6) can also be included in either (7) or (8).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Generalization of D(k, q) and Gn(q)
Let [0]x denote the vector [l] satisfying l∗ = x and lα = 0for α ∈ T , and 〈0〉x denote the vector 〈r〉 satisfying r∗ = xand rα = 0 for α ∈ T .
Clearly, for any T the bipartite graph Γ(T, q) is q-regularand contains the edge ([0]0, 〈0〉0).
For k ≥ 2, let Uk denote the set consisting of the first k−1elements in the following set
U = {η, 0, 1, 01, 10, 010, 101, 0101, 1010, . . .}. (9)
Then, Γ(Uk, q) is equivalent to D(k, q).For positive integer n, let Wn denote the set of all-zero
sequences of length less than n. Then, Γ(Wn, q) is equivalentto the Wenger graph Gn(q).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Automorphisms of Γ(T, q)
For any binary sequence α, let |α| and w(α) denote itslength and the number of its nonzero bits, respectively.
For x, y ∈ Fq, let λx,y denote the map from L(T ) ∪ R(T )to itself such that, for [l] ∈ L(T ) and 〈r〉 ∈ R(T ),
(λx,y([l]))∗ = xl∗, (λx,y(〈r〉))∗ = yr∗,
and
(λx,y([l]))α = xw(α)+1y|α|−w(α)+1lα,
(λx,y(〈r〉))α = xw(α)+1y|α|−w(α)+1rα.
Theorem
For any x, y ∈ F∗q, λx,y is an automorphism of Γ(T, q).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Automorphisms of Γ(T, q)
For α ∈ T , let ST (α) denote the set of sequences β with
{α01β, α10β} ∩ T 6= ∅.
Let S(T ) = ∪α∈TST (α).
Theorem
For any x ∈ Fq and α ∈ T with ST (α) ⊂ T , there is anautomorphism θx,α of Γ(T, q) such that, for any [l] ∈ L(T )and 〈r〉 ∈ R(T ),
(θx,α([l]))α = lα + x, (θx,α(〈r〉))α = rα − x,
and
(θx,α([l]))γ = lγ , (θx,α(〈r〉))γ = rγ ,
for any γ 6∈ {αβ : β ∈ {0, 1}∗}.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Automorphisms of Γ(T, q)
For σ ∈ {0, 1}∗, let σ0 = η and σi = σi−1σ for i > 0.
Theorem
If S(T ) ⊂ T , then for any edge ([l], 〈r〉) there are automor-phisms θ0, θ1 of Γ(T, q) such that
1 θ0([l]) = [0]l∗, (θ0(〈r〉))∗ = r∗ and
(θ0(〈r〉))1i = li+1∗ r∗, if 1i ∈ T, i ≥ 0,
(θ0(〈r〉))α = 0 for all α ∈ T \ {1i : i ≥ 0},
2 θ1(〈r〉) = 〈0〉r∗, (θ1([l]))∗ = l∗ and
(θ1([l]))0i = ri+1∗ l∗, if 0i ∈ T, i ≥ 0,
(θ1([l]))α = 0 for all α ∈ T \ {0i : i ≥ 0}.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Automorphisms of Γ(T, q)
For a ∈ {0, 1}, let Ha(T ) denote the set of sequences obtainedfrom the sequences α ∈ T \ {η} by deleting a a either fromthe leftmost or from two consecutive a’s.
Theorem
1 If H0(T ) ⊂ T , then, for any x ∈ Fq, there is an auto-morphism φ of Γ(T, q) such that
(φ([l]))∗ = l∗, for [l] ∈ L(T ),
(φ(〈r〉))∗ = r∗ + x, for 〈r〉 ∈ R(T ).
2 If H1(T ) ⊂ T , then, for any x ∈ Fq, there is an auto-morphism ψ of Γ(T, q) such that
(ψ([l]))∗ = l∗ + x, for [l] ∈ L(T ),
(ψ(〈r〉))∗ = r∗, for 〈r〉 ∈ R(T ).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Automorphisms of Γ(T, q)
Theorem1 If H0(T ) ∪ S(T ) ⊂ T , then for any 〈r〉 ∈ R(T ) there is
an automorphism π of Γ(T, q) such that π(〈r〉) = 〈0〉0.
2 If H1(T )∪ S(T ) ⊂ T , then for any [l] ∈ L(T ) there is anautomorphism π of Γ(T, q) such that π([l]) = [0]0.
Theorem
If H0(T )∪H1(T ) ⊂ T , then the bipartite graph Γ(T, q) is edge-transitive, or equivalently, for any edge ([l], 〈r〉) ∈ E(Γ(T, q))there is an automorphism π of Γ(T, q) such that π([l]) = [0]0and π(〈r〉) = 〈0〉0.
Corollary
The bipartite graphs D(k, q) and Gk(q) are edge-transitive.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Example 1
For example, let T1 denote the following set
{η, 1, 0, 10, 02, 01, 101, 021, 010, (10)2, 0210, (01)2, (10)21,
0(01)2, (01)20, (10)3, 02(10)2, (01)3, 1(01)3, 0(01)3}
Then, the bipartite graph Γ(T1, q) is edge-transitive.If the sequences in T1 are mapped into {1, 2, . . . , 20} in
order and the symbol ∗ is mapped to 0, then [l] and 〈r〉 areadjacent in Γ(T1, q) if and only if
lk + rk = l0rk−1, k = 1, 2,
lj + rj = r0lj−2, j = 3, 4, 5,
and, for 6 ≤ i ≤ 20,
li + ri =
{l0ri−3, if i ≡ 0 or 1 or 2(mod6),r0li−3, if i ≡ 3 or 4 or 5(mod6).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Invariants for Components of Γ(T, q)
For any x ∈ Fq, let x0 be the multiplicative unit of Fq.
Lemma
For any ([l], 〈r〉) ∈ E(Γ(T, q)), α, β ∈ {0, 1}∗∪{∗} and s ≥ 0,
1 If {α10s, β10s} ⊂ T , then
lα10srβ − rαlβ10s =
s∑t=0
rs−t∗ (rαrβ10t − rα10trβ).
2 If {α01s, β01s} ⊂ T , then
rα01s lβ − lαrβ01s =
s∑t=0
ls−t∗ (lαlβ01t − lα01t lβ).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Invariants for Components of Γ(T, q)
Theorem
Suppose α ∈ {0, 1}∗ ∪ {∗}.1 If α0q is a sequence in T , then
lα0q − lα0 = rα0 − (rα0q + r∗rα0q−1 + · · ·+ rq−1∗ rα0)
is an invariant of Γ(T, q).
2 If α1q is a sequence in T , then
rα1q − rα1 = lα1 − (lα1q + l∗lα1q−1 + · · ·+ lq−1∗ lα1)
is an invariant of Γ(T, q).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Invariants for Components of Γ(T, q)
For any sequence s = (s1, s2, . . .) of nonnegative integers, letµ0(s) = ∗ and, for i ≥ 1,
µi(s) =
{µi−1(s)10si , if i is odd,µi−1(s)01si , if i is even.
Theorem
If α = µ(s1, . . . , s2n+1) is not equal to α = µ(s2n+1, . . . , s1),then for any edge ([l], 〈r〉) of Γ(T, q),
lα − lα +
n∑i=1
s2i∑t=0
ls2i−t∗ (lµ′2i lµ2i−101t − lµ′2i01t lµ2i−1),
=rα − rα +
n∑i=0
s2i+1∑t=0
rs2i+1−t∗ (rµ′2i+110
trµ2i − rµ′2i+1rµ2i10t),
where µi = µi(s1, . . . , s2n+1) and µ′i = µ2n+1−i(s2n+1, . . . , s1).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Invariants for Components of Γ(T, q)
For any sequence s = (s1, s2, . . .) of nonnegative integers, letν0(s) = ∗ and, for i ≥ 1,
νi(s) =
{νi−1(s)01si , if i is odd,νi−1(s)10si , if i is even.
Theorem
If α = ν(s1, . . . , s2n+1) is not equal to α = ν(s2n+1, . . . , s1),then for any edge ([l], 〈r〉) of Γ(T, q),
lα − lα +
n∑i=0
s2i+1∑t=0
ls2i+1−t∗ (lν′2i+1
lν2i01t − lν′2i+101t lν2i)
=rα − rα +
n∑i=1
s2i∑t=0
rs2i−t∗ (rν′2i10trν2i−1 − rν′2irν2i−110t),
where νi = νi(s1, . . . , s2n+1) and ν ′i = ν2n+1−i(s2n+1, . . . , s1).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Invariants for Components of Γ(T, q)
Theorem
If α = µ(s1, . . . , s2n) is not equal to α = ν(s2n, . . . , s1), thenfor any two adjacent vertices [l], 〈r〉 of Γ(T, q),
lα − lα +
n∑i=1
s2i∑t=0
ls2i−t∗ (lν′′2i lµ2i−101t − lν′′2i01t lµ2i−1)
=rα − rα +
n−1∑i=0
s2i+1∑t=0
rs2i+1−t∗ (rν′′2i+110
trµ2i − rν′′2i+1rµ2i10t),
where µi = µi(s1, . . . , s2n) and ν ′′i = µ2n−i(s2n, . . . , s1).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Lower Bounds of the Number of Components
For any binary sequence α = a1a2 · · · an, ai ∈ {0, 1}, letα = anan−1 · · · a1.
Theorem
Γ(T, q) has at least qh connected components, where h is thenumber of pairs α, α ∈ T with α 6= α.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Projection from Γ(T, q) to Γ(T ′, q)
Hereafter, we suppose that T ′ is a subtree of T with η ∈ T ′.Let ΠT/T ′ denote the projection from Γ(T, q) to Γ(T ′, q)
defined naturally.
Theorem
For any component C of Γ(T, q), ΠT/T ′(C) is a component ofΓ(T ′, q) and ΠT/T ′ is a t-to-1 graph homomorphism from C
to ΠT/T ′(C) for some t with 1 ≤ t ≤ q|T |−|T ′|.
� �� �
C ′ = ΠT/T ′(C)
� �� �
CAAAUAAA
? �����
AAAUAAA
? �����
u u′
v1 v′1
vt v′tΠT/T ′
where {v1, . . . , vt} = {v ∈ V (C) : ΠT/T ′(v) = u}.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Lifting to a Component of Γ(T, q)
Assume T ′ = T \ {α} for some leaf node α in the tree T .Let C ′ be a component of Γ(T ′, q) and L(C ′) the set of com-ponents C of Γ(T, q) with C ′ = ΠT/T ′(C). For C ∈ L(C ′)and vertex u ∈ V (C ′), let
ρ(u,C) = {vα|v ∈ V (C),ΠT/T ′(v) = u}.
Then, ρ(u,C) is a coset of some additive subgroup G of Fq.
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ΠT/T ′
C ′
C
L(C ′)
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Structure of the Lifting Sets
Theorem
Suppose T ′ = T \ {α} for some leaf node α in the tree T .Let C ′ be a component of Γ(T ′, q) and C ∈ L(C ′). Then,n = |L(C ′)| divides q and there are maps f : V (C ′) → Fq,g : L(C ′)→ Fq and an additive subgroup G = G(C ′) of Fq oforder t = q/n such that, for C ∈ L(C ′),
ρ(u,C) =
{f(u) + g(C) +G, if u ∈ V (C ′) ∩ L(T ′),f(u)− g(C) +G, if u ∈ V (C ′) ∩R(T ′),
where {g(C)|C ∈ L(C ′)} is a representive set of cosets of Gin Fq, namely, {g(C) +G}C∈L(C′) are distinct cosets of G inFq with
⋃C∈L(C′) (g(C) +G) = Fq.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Projections and Lifts
Theorem
Suppose S(T ) ⊂ T and S(T ′) ⊂ T ′.1 There is an integer t with 1 ≤ t ≤ q|T |−|T
′| such that,for any component C of Γ(T, q), ΠT/T ′ is a t-to-1 graphhomomorphism from C to ΠT/T ′(C).
2 If T ′ = T \ {α} for some leaf node α of T , there is anadditive subgroup G of Fq such that, for any componentC of Γ(T, q) and vertex u of ΠT/T ′(C), the set ρ(u,C) isa coset of G.
Corollary
If S(T ) ⊂ T , all components of Γ(T, q) are of the same size.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Lower Bounds for the Girth of Γ(T, q)
For a ∈ {0, 1}, let Ma denote the set of the sequences β ∈ Uthat are lead by a, namely,
Ma =
{{0, 01, 010, 0101, . . .}, if a = 0,{1, 10, 101, 1010, . . .}, if a = 1.
Lemma
Let β be a sequence in T ∩ U .
1 If β ∈M0, then Γ(T, q) has no cycle of length 2(|β|+ 1)containing a vertex of form [0]x.
2 If β ∈M1, then Γ(T, q) has no cycle of length 2(|β|+ 1)containing a vertex of form 〈0〉x.
Theorem
Assume that S(T ) ⊂ T . If β ∈ T ∩U , then the girth of Γ(T, q)is at least 2(|β|+ 2).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Example 2
For example, for n > 0 let
Xn =
{U4k−1 ∪ {(01)k−10, (01)k0}, if n = 2k − 1,U4k+1 ∪ {(01)k, (01)k+1}, if n = 2k.
From S(Xn) ⊂ Xn, we see that the girth of Γ(Xn, q) is at least2((n+2)+2) = 2n+8. Note that this lower bound is equal tothe best known one achieved by D(2n+ 3, q) ∼= Γ(U2n+3, q).
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1 1 1 10 0 0
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Lower Bounds for the Girth of Γ(T, q)
Lemma
Let s, t be nonnegative integers with gcd(s− t, q − 1) = 1.
1 If β ∈ M0 ∪ {η} and {0sβ, 0tβ} ⊂ T , then Γ(T, q) hasno cycle of length 2(|β|+ 2) containing a vertex of form[0]x.
2 If β ∈ M1 ∪ {η} and {1sβ, 1tβ} ⊂ T , then Γ(T, q) hasno cycle of length 2(|β|+ 2) containing a vertex of form〈0〉x.
Theorem
Suppose S(T ) ⊂ T , a ∈ {0, 1} and β ∈Ma ∪ {η}. If there arenonnegative integers s, t with gcd(s − t, q − 1) = 1 such that{asβ, atβ} ⊂ T , then the girth of Γ(T, q) is at least 2(|β|+3).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Example 3
Let T3 denote the following set
{η, 0, 1, 01, 10, 010, 0101, 01010, 02, 021, 0210, 02101, 021010}.
From S(T3)∪{01010, 001010} ⊂ T3, the girth of Γ(T3, q) is atleast 2(5 + 3) = 16.
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Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Lower Bounds for the Girth of Γ(T, q)
Lemma
1 If β ∈ M0 ∪ {η} and 1mβ ∈ T for some m > 0, thenΓ(T, q) has no cycle of length 2(|β| + 3) containing thevertex [0]0.
2 If β ∈ M1 ∪ {η} and 0mβ ∈ T for some m > 0, thenΓ(T, q) has no cycle of length 2(|β| + 3) containing thevertex 〈0〉0.
Theorem
Suppose a ∈ {0, 1} and S(T ) ∪Ha(T ) ⊂ T . If there are posi-tive integers t, n1, n2, . . . , nt,m and sequences γ, β ∈Ma∪{η}with |β| = |γ|+ 2t− 1 such that {aan1aan2 · · · aantγ, amβ} ⊂T , where a is the symbol in {0, 1} other than a, then the girthof Γ(T, q) is at least 2(|β|+ 4).
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Example 4
Let T4 denote the set of the following sequences
10310, 1031, 103, 102, 10, 1;
02101, 0210, 021, 02, 0; 01; 0310, 031, 03; η.
From S(T4) ∪ H1(T4) ∪ {10310, 02101} ⊂ T4, the girth ofΓ(T4, q) is at least 2(3 + 4) = 14.
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η
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Lower Bounds for the Girth
Theorem
Assume that H0(T ) ∪ H1(T ) ⊂ T . If α ∈ T ∩ U , then thegirth of Γ(T, q) is at least 2(|α|+ 3).
Corollary
For k ≥ 2, the girth of D(k, q) is at least k + 4.
Background
GeneralizedGraphsΓ(T, q)
Automorphismsof Γ(T, q)
Connectivityof Γ(T, q)
Projectionsand Lifts inΓ(T, q)
Girth ofΓ(T, q)
Conclusion
Conclusion
In this talk, we deal with a generalization Γ(T, q) of thebipartite graph D(k, q), by indexing the entries of vertex vec-tors with the nodes in a binary tree T .
1 Sufficient conditions for Γ(T, q) to admit a variety of au-tomorphisms are proposed. A sufficient condition forΓ(T, q) to be edge-transitive is shown.
2 For Γ(T, q), we show some invariants over the connect-ed components. A lower bound for the number of itscomponents is given.
3 Projections and lifts of Γ(T, q) are investigated.
4 A few lower bounds for the girth of Γ(T, q) are deduced.New families of graphs with large girth in the sense ofBiggs can be obtained from Γ(T, q), such as Γ(Xn, q).